# Understanding the Camera Numbers -Aperture f-stop, Shutter Speed, ISO

This article is about "Understanding the camera Numbers". It is NOT a primer about using those numbers to take better photos (for that, see for example).

## Where the Numbers Come From

The number for lens f/stop in photography (for example, f/8) is the ratio of lens focal length divided by the effective lens aperture.

That is math and physics, but still the very useful purpose of f/stop numbers is the grand concept that the same f/stop, like say f/8, will be the same exposure in any lens, regardless of focal length, or physical size of construction.

Is f/stop written f/stop or f-stop or fstop? The lens manufacturers properly write f/8. The internet changes things, and the term f-stop has become very popular on-line, but we also still see f/stop. I learned to write f/stop, because we also write f/8, and it reminds me of the division defining it, however it is debated today that it surely refers to the "fractional system" originating it (new in 1895).

f/stop number = focal length / aperture diameter

f/8 is an aperture diameter of literally = focal length / 8. This is a common and useful approximation, the actual physics is deeper (look up "numerical aperture" and "conservation of etendue"). The "aperture diameter" is the diameter of the entrance pupil from in front of the lens. But what photographers need to know is that the purpose of using the f/stop system is so that the same numerical f/stop on any lens is expected to produce the same exposure.

OK, yes, there can be small minor variations, especially in the old days in lenses without modern coatings. Reflection losses on uncoated glass can be 4% or more at each glass surface, but best modern coatings can reduce this to maybe 0.1%. Lens coatings cancel the reflection, allowing the light to pass through the lens, instead of reflecting it away. Fancy lenses today may contain 12 or 16 glass groups (24 or 32 surfaces), each surface losing a slight amount of light even if with best coatings, and so suffer more. But lenses normally have only slight variations today, but which are still important in professional movie cameras, when switching lenses on the same scene, or mixing scenes shot on different days.

T-stops: So the professional movie lenses use T-stops, with markings which are calibrated to the actual amount of light the lens transmits, instead of the theoretical amount (a T2 lens actually transmits the light that a perfect f/2 lens should, matching the light meter).

Modern lenses today use much improved coatings to improve this tremendously. So now, this is relatively solved for still cameras, since the camera meters through the lens anyway, automatically accounting for any possible variance in the lens losses.

## A few Basic Definitions about Lenses

Aperture is not exactly the obvious physical diameter (shown by black vertical lines in image above), but instead is seen through the magnification of the front lens elements as the apparent "working" diameter D (named Entrance pupil). The physical aperture diameter is designed accordingly.

Focal length: The marked Focal Length number applies when focused at infinity. Focal length changes as we focus closer (focal length normally becomes longer if front elements are extended to focus closer). This change is relatively minor if at focus distances of a few feet or more, but at 1:1 macro, the focal length becomes 2x marked, and equal to the working distance in front of the lens (specifically, to the front Principle Point H). The actual focal length is measured to the rear Principle Point, H', as shown above. Due to varying focus distance, the focal length marked for infinity is not always necessarily the exact distance to the sensor plane.

The front Principle Point H is where the vertex of the Angle of View through the Entrance Pupil is located. Same with H' and Exit Pupil. The front Angle of View from H is necessarily the same Angle of View at H' back to the sensor (so, the sensor size and focal length determine the numerical Angle of View). The Principle Points are on the designer's apparent planes where the subject and sensors images appear to be. Design of lens elements can move these points, and both H and H' points are often inside the lens, but in fact, they can often be moved literally outside the actual lens, either in front or behind the lens. They are moved by adding convex lens elements that converge, or concave lenses that diverge. Zoom lenses greatly complicate this.

Wikipedia can show the definition of the Principle Points (H and H') of a lens. In the classic Thin Lens model (which is a hypothetical single element lens, like a simple magnifying glass, or a pinhole), the points H and H' are at the same point in the center of the single element. In an actual multi-element camera lens, these points move apart, with more glass elements (with different index of refraction) used to correct color aberration in the lens, which relay the image between the two points.

Panoramic photos (combining multiple photos) show the most accurate perspective (of close subjects) when the panoramic camera is rotated on an axis through the H point. Panoramic articles describe ways to determine this pivot point they call Nodal point, or sometimes Perspective point.

Here are diagrams of two (old) Nikon prime lenses:

Telephoto - H' is moved to be slightly in front of lens.
In telephoto lenses, the H' point is designed slightly in front of the front lens element, because, the actual optical technical definition of "telephoto" is that the lens is made to be physically shorter than its focal length (which is a practical way to build a smaller long lenses).

"Telephoto" does not actually mean the subject is at long distance, since wide angle lenses focus at infinity too. FWIW, telephoto refers to moving the H' point, to design a camera lens physically shorter than the focal length, which is typically used for long focal lengths, but telescopes use entirely different designs.

Retro-focus wide angle, H' is moved to be behind the lens.

Wide angle lenses are normally retro-focus (for SLR, DSLR), which means the rear node H' is designed well behind the rear element. This allows the short lens to be mounted well forward, leaving space for the SLR camera mirror to be raised. Otherwise for example, an 20 mm lens would block raising a mirror 24 mm tall.

Elements and Groups: Complex camera lenses are designed with several individual glass elements. An element is an individual piece of optical glass, with curves (convex converges the light toward center, or concave diverges the light, seen in the picture at page top). A single element is counted as a group of one, but sometimes two elements are cemented together to combine into one group (specifically, with no air boundary between them). These doublet group elements each have different Index of Refraction, to bend color wavelength differently (to correct color aberration). The wide angle lens at right just above has nine elements in seven groups. That telephoto lens has four elements in three groups.

Field of View: The field of view of the lens is determined by the angle from the rear principle point H' (focal length) back to the sensor size dimensions (sensor size is a major factor of field of view). Then the same angle from the subject principle point H determines the field of view forward to the focus distance. Focal length and sensor size are the factors of angular field of view. This is why sensor size determines "Crop Factor", which is about the view seen by the same lens on different sensor sizes.

Magnification: Binoculars and telescopes (using a magnifying eye-piece lens) describe "x power optical magnification" relative to the size that the bare human eye sees it. The angular size of the full moon is 0.5°. In binoculars with 10x magnification it appears to subtend an angle of 5°. Magnification of a telescope is normally given as (focal length of objective lens / focal length of eyepiece lens).

However, cameras are different, being reproduction devices (and the eye does not see the actual image). In cameras, size can be measured, and magnification is computed relative to the actual size of the original object. The ratio of the subject's image size on the sensor to its real actual life size (image:subject) is called Magnification, which is normally a size reduction. If our Moon is 3474 km diameter, and its image is 0.5 mm, that's a very great size reduction. If a real subject that is physically 100 mm wide is reproduced to be 10 mm wide on the sensor image (on the film, so to speak), then the magnification is 10/100 = 1:10 = 0.1x reproduction ratio (actually a reduction). Which 1:10 also happens to be the same ratio as the distances behind and in front of the lens then, also necessarily 1 to 10.

If at 1:1 reproduction ratio (macro), then of course1 1:1 means equal sizes, in real life and in the camera sensor image. So then at 1:1, the "working distance" in front of the lens (in front of point H) is necessarily equal to the distance behind the lens (the modified focal length, behind point H'), due to similar triangles, etc. Seems a cute fact, which aids understanding, however (today, with zoom lenses and internal focusing which shifts thing inside), we probably are not told the exact locations of H and H' (which change with zoom and focal length).

Back to the camera "numbers":

Lenses will expose equally if set to the same f/stop. That's what f/stop is, and means, and is for. A light meter may tell you to use ISO 100 f/16 at 1/100 second, which does not depend on which lens or camera sensor you might use.

Details of Why for f/stop: For two lenses at the same f/8, the lens with 3x longer focal length has an aperture diameter 3x larger. Tricky, but the 3x focal length magnifies the view 3x, and then crops it to 1/3, so only 1/3x width and 1/3x height is seen, which is 1/9 area, which only reflects 1/9 the light the wider lens sees. But the longer 3x lens also has aperture 3x larger, which is 9x area, and so now admits 9x more light, which before was 1/9 as much, from a 1/9 area field... so the 9x times 1/9 result is the same exposure in both f/8 lenses. Another argument is the Inverse Square Law over the 3x longer focal length is 1/9 the light, when the image reproduction reaches the sensor plane (which is just repeating the first explanation again). This is why the f/stop numbering system is used. It's good stuff. It means that f/8 is f/8 in any lens, producing the same exposure.

So, f/8 denotes (focal length / 8), which represents the aperture of the lens (effective diameter as seen through the front lens element). This exposure value can be compared with other lenses in this way. A series of multiple f/stop steps is designed, called "stops". Stop originally denoted the notched detent which marked the 2x area multiples of f/stop. Today in photography, the word stop is used to mean any step of double or half value of exposure, also in regard to shutter speed and ISO. Each full stop towards larger f/stop numbers gives half the light exposure of the previous step (called stopping down, which also increases depth of field).

The charts on the next page show all the computed camera numbers (f/stop, shutter, and ISO). The table below is the fractional f/stop steps in tenth stops. One purpose could be to aid determining span in stops between two values, but the calculators below do that too.

### Rounding the Nominal Numbers

There are two concepts of camera numbers here. "Nominal" numbers (existing in name only) are the numbers actually marked on the camera, which are just simpler approximations of what I call the "Precise" numbers that the camera actually uses. The camera knows to actually do it right, but the marked numbers are shown, made easier for humans.

Nominal f/stop numerical values marked might be rounded, or might be truncated, but are often approximated into a friendly ballpark number for humans. But to make each stop always be exactly a 2x difference, the lens and camera has to actually use the exact precise values. Specifically, to honor 2x stops, the precise values use the sequence 1,2,4,8,16,32, etc. Also 1/those for shutter speeds. Or f/stops increment in steps of √2 (then every second f/stop is 1,2,4,8,16,32). Third and half stops are intermediate values. Much more on next page, with the charts.

For example, the cameras and light meters are marked f/11, and we say it as f/11, but f/11.314 is the necessary correct actual calculated value. This is only about 0.08 stop difference, and any difference exists only in our mind, since the camera will know to do it right anyway. Most other exposure factors are larger than this difference, and most other f/stop values are closer, but Guide Number calculations for speedlights can use f/11.31 instead of f/11.

As a simple way to show this fact is obviously true (that f/11 is actually f/11.314), note that f/stop numbering is the sequence of √2 intervals, (which are 1.414 numeric intervals) - making every other stop number be a multiple of 2. This sequence of progressions, when arranged into rows of every other doubled aperture values, are:

 f/ 1 2 4 8 16 32 f/ 1.414 2.828 5.657 11.314 22.627

It can be handy to realize that doubling any f/stop number (for example, f/7 to f/14) is exactly two stops. The sequence 1,2,4,8,16 etc are exact precise numbers, but the others are more unwieldy, and are marked with approximate numbers. But the camera knows to do it right.

Shutter speed marking numbers are also similarly approximated. For example, the camera nominal markings show 1/20 second and 1/10 second (and 10 and 20 seconds) to be both third stop values and half stop values. But the same value cannot be both values, and the camera does compute the actual value closer (half stop 20 seconds will be 22.6 seconds, and full stop 30 seconds will be 32 seconds, see standard shutter speed charts on next page). The camera does it right, but we humans are frequently shown easier approximated numbers. Unless we're doing calculations, we really don't care about the precise numbers. But each stop being exactly 2.0x exposure is very important to us.

## Calculator to Compare Difference of any two f/stop or shutter or ISO values

There is also an Exposure calculator to compare two exposures by including all three parameters combined. Or these are individual calculators here.

It is not obvious that the difference between f/4 and f/5 is 2/3 stop, and f/9 to f/10 is 1/3 stop, so the calculator purpose is to help with the math.

The selections below provide the possible settings (of full, third, half stops).
Full stops are Green. Third stops are Blue. Half stops are Red, also flagged with .

• Option 1 selects the marked camera nominal numbers, but like the camera, it uses the corresponding precise values that the camera actually uses (better accuracy than the nominal marked numbers). The next page has a chart of the actual precise values that the camera uses, and the calculator matches that here.
• Option 2 can be any numbers. Comparing results of calculated f/stop values would use this, however otherwise, not sure there is much use for this option, since the only numbers we can select on the camera are the marked nominal numbers (hence Option 1 is more useful and accurate). But for reasons stated, in Option 2, f/8 to f/11, or from 1/125 to 1/60 second, is NOT exactly 1.000 EV. Option 1, it is. But again, the camera knows to do it right.
• Option 3 assumes that adding +EV exposure adds more shutter time, or causes a lower and wider f/stop number, or a higher ISO number (like the camera does).
The EV increment can be any numeric value, but of course 1/3 stop is 0.3333 EV, 2/3 stop is 0.6667 EV, and 1/2 stop is 0.5 EV (or better, you can enter 1/3, 1/2, or 2/3 there, or 1/6 or 1/10, etc, which are calculated precisely).

The calculator range is large, but not quite infinite.
f/stop range is f/0.5 to f/520, 20 EV (f/0.5 is a theoretical limit for the lens to still focus in air).
Shutter speed range is 1/32000 to 512 seconds, 24 EV.
ISO is from 1.56 to 819200, 19 EV. Seems adequate.
Range here just meaning, it can compute further, but the suggested nearest third nominals hold at those limits.

• Option 4 tenth f/stop values are also detailed in the tenths chart just below, but this option is the difference from tenth stop values to a nominal value we can set on the camera. Difference will not exceed 0.17 EV from the nearest third.

## f/stop CalculatorStops Difference

1. Precise values of nominal f/stop values
(full, third, half stops)

from A:    to B:

2. Difference in any two numeric values

from A:  f/   to B:  f/

3. Add ± EV exposure to a f/stop value

Formats can be ± 1, 2.333, 2/3, 1 1/2

4. Tenth stops to Nominal stop values

from plus   to

## Shutter Speed CalculatorStops Difference

1. Precise values of nominal shutter values
(full, third, half stops)

A:    to B:

2. Difference in any two numeric values

from A:    to B:  second

3. Add ± EV exposure to a shutter value

Formats can be ± 1, 2.333, 2/3 or 1 1/2

## ISO CalculatorStops Difference

1. Precise values of nominal ISO values
(full, third, half stops)

A:  ISO   to B:  ISO

2. Difference in any two numeric values

from A:  ISO   to B:  ISO

3. Add ± EV exposure to ISO value

Formats can be ± 1, 2.333, 2/3, 1 1/2

Please report ( Here ) any problems with the calculator, or with any aspect of this or any page. It will be appreciated, thank you.

## Tenth Stop Usage

Cameras today can use third stops. If using a modern camera, I cannot imagine any reason not to choose third stops. Greater precision of adjustment can only be a good thing. When the light meter reads a certain value, the camera can be set to the nearest third stop. The error is not more than 1/6 stop (half a third). So the calculators above try to flag the 1/2 stop values as possibly unintended.

But handheld light meters typically can also be set to read tenth stops (for metering multiple flash). If you set your light meter to read in tenth stops, the format of the result value we see is (for example):

f/8 plus 7/10 stop

If reading EV, this is EV 8.7. But if for f/stops, this is NOT f/8.7. It is 7/10 of the way between f/8 and f/11 - or about f/10, but read as "f/8 plus 7/10 stop".

By definition, the equivalent value of f/8 plus 7/10 stop is simply two third-clicks past f/8, or one third-click below f/11 (easy to set). The camera dial will indicate f/10 there, but we can instead meter and work in tenth-stop differences from full stops.

Fractions: 1/10 is 0.1 stop. The fraction 1/3 stop is 0.333 stops, and 2/3 stop is 0.667 stops, so a reading around 0.3 is one third stop, and one around 0.7 is two third stops. The lens can only be set to third stops, so just pick the nearest third stop: 0, 1/3, 2/3, or 1 stop.

There would not seem much point of 1/10 stop meter readings for daylight (IMO), since we can only set the camera to the nearest third stop. Maybe metering in tenths could give us an indication that the 1/3 stop exposure we set was actually a bit more or less than metered.

However there are two very good reasons to use tenth stops for metering multiple flash. One is for greater precision in adjusting the power level of individual flash units - the actual difference between two lights could be controlled more closely. But the overwhelming advantage is when pondering fill level for that lighting ratio - how much is one and a third stop less than f/10? It is about f/6.3, but who knows that? But if we read these two values as f/5.6 plus 3/10 stop vs. f/8 plus 6/10 stop, then in our heads we easily know ratio is 1.3 stops difference, immediately (in use, that is really a big deal).

Change precision shown below

## F/stop Table of Tenth-Stops

Notes:   f/stop = √2 ^(stop number + fraction)   (√2 is 1.4142)

e.g., 2/10 stops past f/11 (stop number 7) is √2 7.2 = f/12.126
Or 1/3 stop past f/11 is √2 7.3333 = f/12.698

If interested, here is a one page printable PDF file of this tenths chart.

Nominal and precise values of all (full and third and half) stops are on the next page.

## How the Stops are Numbered (Origin of these numbers)

### f/stop

The purposes of f/stop nomenclature:
• Prior to the mid-1920s, Kodak used the "U.S. system" (Uniform System, from Britain in 1880s) of aperture markings, where stops were stop numbers 1, 2, 4, 8, 16 ... with U.S. 1 starting at todays f/4 (and 2 corresponded to f/5.6, and 3 to f/8, etc). And there were other systems back then (Zeiss, etc), but at any number in any system, different focal length lenses gave different exposure results. Two photographers standing at the same scene with different lenses probably could not use the same exposure numbers.

The reason to invent and use the f/stop method of numbering aperture is so that any two lenses (of any different sizes or focal lengths), will give the same exposure if at the same numerical f/stop. The entire idea is that f/4 exposure is f/4 exposure, in any lens of any size. This way:

The focal length affects the magnification of the field of view. Doubling the focal length zooms in to half of the view width.
A short lens (wide angle) gathers a lot of light from a wide view, and concentrates that light onto the camera sensor area.
A long lens gathers less light from a smaller view, onto the same sensor area.
But fstop = focal length / aperture diameter equalizes these, a larger aperture in a longer lens, giving equal exposure at equal f/stop numbers. Exposure is about Illumination per unit of scene area, which stays the same. That's why we bother with f/stop numbers, the benefit is great. f/8 is always f/8, on any lens. Our light meter works the same for any lens.

John A. Hodges championed this new "fractional numbering system" in 1895, which was eventually universally adopted.

f/stop number = focal length / aperture diameter

• Values of f/stop intervals are selected to provide useful standard steps, called stops, each stop providing 2x or /2 of the previous step's light. An aperture with double or half area will pass double or half the light. This way:

Aperture is circular, and the area of a circle is defined as Pi r². Double area is twice the light, or one stop.
For double area: 2 Pi r² = Pi (1.414 x r)² , so 1.414x radius gives one stop. √2 is 1.414.
Since f/stop = focal length / aperture diameter, then f/stop numbers increase in 1.414x steps (or 1/1.414 is 0.707x decreasing steps).
Inversely, when the diameter and area are made larger, the f/number from the ratio f/d becomes a smaller number.

Cameras today also have 1/3 stop or 1/2 stop increments. See the note under the tenth stop table above for a formula computing f/stop for partial stops. See next page for an implementation.

• F/stop numbers as f/d are numbered in powers of √2, which is convenient in the math. For any Equivalent Exposure, f/stop number squared x shutter speed is the EV number of the exposure.

Full f/stop numbers advance in steps of 1.414x numeric multiples (f/1, f/1.4, f/2, f/2.8, f/4 ...) From any f/stop number, in all cases, double or half of that number is two stops (for example, f/10.2 is two stops above f/5.1). Every second stop is the doubled f/number. Or one stop is x1.414 (or /1.414 which is x0.707).

Third f/stop numbers advance in multiples of the cube root of √2, or 1.12246x the previous (speaking of f/stops).
Every three third-stop steps (from any point) is exactly 1.0 stop and a 2.0x change of the light.

Half f/stop numbers advance in multiples of the square root of √2, or 1.1892x the previous (speaking of f/stops).
Every two half-stop steps (from any point) is exactly 1.0 stop and a 2.0x change of the light.

Less is more, Less f/stop number is More light.

Lens manufacturers seem to truncate numbers instead of round off. For example, f/5.6 is actually 5.66, and f/3.5 is 3.56. Except we see the same f/1.2 marking for the half stop (f/1.189) and third stop (f/1.260). Point is, the markings are just nominal numbers to show us humans. The lens and camera know to try to do it right.

### Shutter speed and ISO

The values of shutter speed and ISO are linear scales (square root of 2 is Not involved), meaning that 2x the number is a 2x difference, and 2x is one stop. The very important thing to the definition of our exposure system is that any span of three third stop steps (or any two half stop steps) must come out exactly 1.0 stop of 2.0x exposure difference. To force this, cube root (and square root) steps are the proper values to create and number step intervals.

The next third-stop shutter step is cube root of 2 (1.26992) greater than the previous value (but for f/stop, see above)
Every three third-stop steps (from any point) is exactly 1.0 stop and a 2.0x change of the light.

The next half-stop shutter step is square root of 2 (1.4142) greater than the previous value.
Every two half-stop steps (from any point) is exactly 1.0 stop and a 2.0x change of the light.

The next full-stop value is 2x greater than the previous value. Doubling any numeric value is one stop (speaking of shutter speed or ISO, but 2x number is two stops for f/stops, see above.)

Again (f/stop, shutter speed, ISO), the nominally marked numbers may not be the exact precise values, but the camera knows exactly what to do.

For example, set ISO 250 or ISO 2000 in the Nikon DSLR camera. Then near the top of the Exif data will show the ISO 250 or 2000 values, but farther down in the manufacturers data, it shows the precise values used, ISO 252 or ISO 2016. (The ISO base is obviously 100 today, instead of 1... 100, 200, 400, 800 instead of 1, 2, 4, 8 (otherwise, ISO 100 would a third stop less than ISO 128). This makes third stops of 252 and 2016 instead of full stops 256 and 2048 - which we call 250 and 2000.) Auto ISO is probably using 1/6 stops, but which will be steps of the precise sixth root of 2. The numbers we see are just convenient nominal numbers, which the number really does not much matter to us humans. We just want one stop to always be a 2x light value. The point here is that the camera typically uses numbers a little different than the numbers we see. The only time that actually matters is if we try calculating ourself, using the nominal numbers instead of the actual precise numbers.

 Marked Shutter Speed Precise 30" 30 seconds 32.0 sec. 2" 2 seconds 2.0 sec. 2 1/2 second 0.5 sec. 30 1/30 second 1/32 sec. 1000 1/1000 second 1/1024 sec. See next page charts

Shutter speed is of course the time duration when the shutter is open, exposing the sensor or film to the light from the aperture. On many cameras, numerical values for shutter speed are marked on the camera using two methods with different meanings - for example, marked as either 30 or 30". Just the number alone, like 30, is an implied fraction (1 over the number), meaning 1/30 second. The same number written 30" means 30 whole seconds, not a fraction. However, these are nominal markings, and 1/30 nominal is actually precisely 1/32 second (precise values run 1,2,4,8,16,32 seconds or 1/ those values). A slow shutter is a longer duration, and a fast shutter is a shorter duration.

A flash, especially a speedlight flash, is typically a much shorter duration than the shutter. The flash simply must occur while the shutter is open (sync), but the faster flash exposure is not affected by the slower shutter speed. Keeping the shutter open longer does increase the continuous ambient light seen, but shutter speed does not change what the fast flash does.

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